Distribution problems for L-functions and number fields

L 函数和数域的分布问题

• 批准号: RGPIN-2021-02952
• 负责人: Akbary, Amir
• 金额: $ 1.53万
• 依托单位: University of Lethbridge
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=742292
• 关键词: Distribution; problems; functions; number; fields

This research program is in the field of number theory, which is the part of mathematics that deals with properties of integers and solutions of equations with integer coefficients. Such equations are called Diophantine equations. Elliptic curves are one of the fundamental objects of study in the theory of Diophantine equations. Many number theoretical problems can be studied more generally in certain subsets of complex numbers called number fields. Over the past decade, I have studied several number theoretical problems related to elliptic curves and number fields from an analytic point of view and by employing L-functions which are fundamental tools in analytic number theory. I plan to further my research in such problems over the next five years. I aim to extend some of the fundamental results of analytic number theory, as originally stated for the Riemann zeta function and Dirichlet L-functions, to the general setting of automorphic L-functions; to prove new value distribution theorems for certain families of L-functions and explore the applications of such results in the distribution of invariants of number fields such as Euler-Kronecker constants; and to investigate distribution problems arising in geometric settings formulated in terms of division fields of elliptic curves and their associated L-functions. The outcome of this research will enrich our knowledge in several key areas of number theory and will help to pave the way for further studies in the field. In addition, this research program provides excellent training opportunities for students and postdoctoral fellows.

这个研究项目是在数论领域，这是数学的一部分，涉及整数的性质和整数系数方程的解。这样的方程称为丢番图方程。椭圆曲线是丢番图方程理论的基本研究对象之一。许多数论问题可以在称为数域的复数的某些子集中进行更一般的研究。在过去的十年中，我研究了几个数论问题有关的椭圆曲线和数域从分析的角度来看，并采用L-功能，这是基本的工具，在解析数论。我计划在未来五年内进一步研究这些问题。我的目标是将解析数论的一些基本结果，如最初对Riemann zeta函数和Dirichlet L-函数所述，推广到自守L-函数的一般情况;证明某些L-函数族的新的值分布定理，并探索这些结果在数域不变量分布中的应用，如Euler-Kronecker常数;并研究在椭圆曲线及其相关L-函数的划分领域中制定的几何设置中出现的分布问题。这项研究的成果将丰富我们在数论几个关键领域的知识，并将有助于为该领域的进一步研究铺平道路。此外，该研究计划为学生和博士后研究员提供了极好的培训机会。